Lecture Event

**SFB Colloquium Representation Stability (Research Project C1 and C3)**

Prof. Benson Farb (University of Chicago)

7 Jan 2014, 16:00 – 18:00

Talk

**Representation stability: a user's guide**

Prof. Benson Farb (University of Chicago)

7 Jan 2014, 16:00 – 16:45

``Representation stability'' refers to a phenomenon discovered a few years ago by Church-Farb that seems to occur all over mathematics; it was developed into a powerful theory with Ellenberg. One simple application gives results such as: the sequence of vector spaces $V_n$ has dimension equal to a polynomial $P(n)$ for $n$ large enough. A common application is to the fixed degree (co)homology of a sequence of spaces $X_n$.

This has been applied to examples in algebraic topology (configuration spaces), algebraic geometry (moduli spaces of surfaces with n marked points, spaces of polynomials on rank varieties), number theory (cohomology of congruence subgroups), algebraic combinatorics (co-invariant algebras), and several other areas. In most cases nothing is known about the actual dimension of $V_n$, but this is now reduced in principle to a finite problem.

The purpose of this talk will be explain to workers in different areas what this theory can do for them, and how they can apply it.